What I did in a 1960's end of terrace three story house without a cavity where the staircases/hallway ran all the way up from bottom to top, was to internally fit plasterboard with polystyrene and aluminium foil attached on the inside.
In made the wall about 1" thicker on the inside It took about 2 days to fit, skim and then paint.
Far easier and cheaper to do than fill cavities that were not designed to be filled, and you won't get damp issues
End result is invisible, but bought up the temperature in the hallway by about 5 degrees.
Something like this stuff. (But I'm sure we did not pay even half the price per sheet they are quoting),
https://www.travisperkins.co.uk/Knauf-Thermal-Laminate-Plus-Tapered-Edge-Plasterboard-35mm-x-2400mm-x-1200mm-(2-88m²-Sheet)/p/603575